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# Kumar-Langmuir¶

This extension of the Langmuir isotherm (see Section Multi Component Langmuir) developed in [11] was used to model charge variants of monoclonal antibodies in ion-exchange chromatography. A non-binding salt component $$c_{p,0}$$ is added to modulate the ad- and desorption process.

\begin{aligned} \frac{\mathrm{d} q_i}{\mathrm{d} t} &= k_{a,i} \exp\left( \frac{k_{\text{act},i}}{T} \right) c_{p,i} q_{\text{max},i} \left( 1 - \sum_{j=1}^{N_{\text{comp}} - 1} \frac{q_j}{q_{\text{max},j}} \right) - k_{d,i} \left( c_{p,0} \right)^{\nu_i} q_i && i = 1, \dots, N_{\text{comp}} - 1 \end{aligned}

In this model, the true adsorption rate $$k_{a,i,\text{true}}$$ is governed by the Arrhenius law in order to take temperature into account

\begin{aligned} k_{a,i,\text{true}} = k_{a,i} \exp\left( \frac{k_{\text{act},i}}{T} \right). \end{aligned}

Here, $$k_{a,i}$$ is the frequency or pre-exponential factor, $$k_{\text{act},i} = E / R$$ is the activation temperature ($$E$$ denotes the activation energy and $$R$$ the Boltzmann gas constant), and $$T$$ is the temperature. The characteristic charge $$\nu$$ of the protein is taken into account by the power law.